March attention has been paid to the multipartite
Greenberger-Horne-Zeilinger (GHZ) state
\(|GHZ\rangle =\frac{1}{\sqrt{2}}(|000\rangle +|111\rangle )\)  or the
generalized GHZ states under the Hawking radiation. It is recently
found in Refs. [1]} and [2]} that the GTN is not
redistributable, but GTE can be redistributed through Hawking
effect. With the growth of the Hawking temperature, the physically
accessible GTN decreases, but physically inaccessible GTN is not
generated. The GTE however behaves differently: with the loss of the
physically accessible GTE, the physically inaccessible GTE is
generated continually through Hawking effect. This is inconsistent
with intuition since quantum entanglement and quantum nonlocality
are in consistent for pure states [3]}, [4]}, [5]}. To clarify
such problems, in this article we study the GTN of multipartite
mixed states for Dirac fields in the background of a Schwarzschild
black hole. The paper is organized as follows. In the second
section, we briefly recall the quantization of Dirac fields in the
background of Schwarzschild black hole, and the GTN, GTE and quantum
coherence for three-qubit X-states. In the third section, we
calculate and analyze the GTN of a detailed mixed state after one or
bipartite acceleration. A novel phenomenon that is basically
different from that of pure states is illustrated: the Hawking
effect may not only degrades the physically accessible GTN, but also
enhances the physically inaccessible GTN. The “sudden death" of
physically accessible GTN takes place at some critical Hawking
temperature, while “sudden birth" happens to the physically
inaccessible GTN. We show that such phenomenon may be not observed
for other mixed states. In the fourth section, we consider the
quantum correlation GTE and quantum coherence. Analytical tradeoff
relations among the distributions of GTE and coherence are derived.
The last section is the conclusions and discussions.
